Question : If $\theta>0$ be an acute angle, then the value of $\theta$ in degrees satisfying $\frac{\cos^2\theta-3 \cos\theta+2}{\sin^2\theta}=1$ is:
Option 1: 90°
Option 2: 30°
Option 3: 45°
Option 4: 60°
Correct Answer: 60°
Solution :
Given:
$\frac{\cos^2\theta-3\cos\theta+2}{\sin^2\theta}=1$
⇒ $\cos^2\theta-3\cos\theta+2=\sin^2\theta$
⇒ $\cos^2\theta-2 \cos\theta+1-\cos\theta+1=\sin^2\theta$
⇒ $(\cos\theta-1)^2-(\cos\theta-1)=1-\cos^2\theta$
⇒ $(\cos\theta-1)(\cos\theta-1-1)=1-\cos^2\theta$
⇒ $(\cos\theta-1)(\cos\theta-2)=-(\cos\theta-1)(1+\cos\theta)$
⇒ $\cos\theta-2=-1-\cos\theta$
⇒ $2\cos\theta=1$
⇒ $\cos\theta=\frac{1}{2}$
⇒ $\theta=60°$
Hence, the correct answer is 60°.
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